A novel flexible exponent power-X family of distributions with applications to COVID-19 mortality rate in Mexico and Canada

This paper aims to introduce a novel family of probability distributions by the well-known method of the T–X family of distributions. The proposed family is called a “Novel Generalized Exponent Power X Family” of distributions. A three-parameters special sub-model of the proposed method is derived and named a “Novel Generalized Exponent Power Weibull” distribution (NGEP-Wei for short). For the proposed family, some statistical properties are derived including the hazard rate function, moments, moment generating function, order statistics, residual life, and reverse residual life. The well-known method of estimation, the maximum likelihood estimation method is used for estimating the model parameters. Besides, a comprehensive Monte Carlo simulation study is conducted to assess the efficacy of this estimation method. Finally, the model selection criterion such as Akaike information criterion (AINC), the correct information criterion (CINC), the Bayesian information criterion (BINC), the Hannan–Quinn information criterion (HQINC), the Cramer–von-Misses (CRMI), and the ANDA (Anderson–Darling) are used for comparison purpose. The comparison of the NGEP-Wei with other rival distributions is made by Two COVID-19 data sets. In terms of performance, we show that the proposed method outperforms the other competing methods included in this study.

Using Eq. ( 2), Marshal and Olkin 4 derived two special sub-models namely, Marshal-Olkin exponential and Marshal-Olkin Weibull distributions.Latterly, the authors used Eq. ( 2), and several probability distributions have been proposed in the literature, see (Ghitany et al. 5 , Gui et al. 6 , and Saboor et al. 7 ) for more reading.
Similarly, in this regard, Mahdavi and Kundu 8 also proposed a new family of distributions by incorporating one additional parameter to the baseline distribution.They named their proposed method by Alpha Power transformation (APTra) family of distributions.The CDF of the APTra family is defined by Using Eq. ( 3), Mahdavi and Kundu 8 modified the exponential distribution and named the alpha power transformed exponential (APTra-Expo) distribution.Furthermore, considering Eq. ( 3), various contributions have been made in the literature on distribution theory; see Dey et al. 9 , Ihtisham et al. 10 , and Hassan et al. 11 .
In the recent past, Shah et al. 12 proposed a new method of probability distribution by incorporating one additional parameter into baseline distribution.Their proposed method is called, the new generalized logarithmic-X (NGLog-X) family of distributions.The CDF of the NGLog-X family is given by Using Eq. ( 4), Shah et al. 12 modified the Weibull distribution and named a new generalized logarithmic Weibull (NGLog-Wei) distribution.For recent developments about the distributional approaches, we refer to a superior extension for the Lomax distribution with application to Covid-19, proposed by Alsuhabi et al. 13 , a novel logarithmic approach to generate new probability distributions, proposed by Zhao et al. 14 , a novel updated-W family of distributions, proposed by Alnssyan et al. 15 , a new Type 1 Alpha Power family of distributions, proposed by Tekle et al. 16 , the Type II-Topp-Leone-Gompertz-G family of distributions with applications to COVID-19 data, proposed by Chamunorwa et al. 17 , a Weighted Cosine-G family of distributions, proposed by Odhah et al. 18 , some inferences on three parameters Birnbaum-Saunders distribution, developed by Shakil et al. 19 , a statistical analysis of excess mortality mean at Covid-19 in 2020-202, proposed by Raihen et al. 20 , exponentiated generalized Weibull exponential distribution, proposed by Abonongo et al. 21, a case study for Kuwait mortality during the consequent waves of COVID-19, derived by BuHamra et al. 22 , and a new inverse Rayleigh distribution with applications of COVID-19 data, developed by El-Sherpieny et al. 23 .
In this research paper, taking motivation from the above discussion, we also propose a new method for obtaining more flexible probability distributions.The proposed method is obtained by implementing the T-X family approach proposed by (Alzaatreh et al. 24 ).The proposed method may be named a Novel Generalized Exponent Power-X (NGEP-X) family of distributions.Based on the NGEP-X method, the improvised version of the Weibull distribution with distributional flexibility in shapes of PDF (probability density function) and HF (hazard function) is introduced.Based on COVID-19 data sets, the fitting power of the proposed work is compared with Alpha Power Transformed Weibull, New Reduce Logarithmic Weibull, Kumaraswamy Weibull, Weibull, Marshal Olkin Nadarajah Haghigh, and Gull Alpha Power Weibull distributions; see Table 3, for references of these competing distributions.
The rest of the work done in the study is organized into seven sections: In section "NGEP-X family", the newly proposed family of distributions is comprehensively derived.Section "NGEP-Wei distribution", gives a special sub-model of the proposed family, named a Novel Exponent power Weibull distribution, and in the same section, the shapes of its CDF, survival function, PDF, and HF are also graphically illustrated.The mathematical properties of the NGEP-X family are given (or derived) in Section "Basic mathematical properties".The method of Maximum Likelihood Estimation is applied for estimating the model parameters of the proposed method in section "Estimation, experiment and simulation".Practical applications via two COVID-19 data sets (describing the mortality rates of the countries of Canada and Mexico) are discussed in section 6.Finally, section 7 gives the "Concluding remarks" based on the analyses done in this paper. (1) Definition Let m(t) = e −αt be the PDF of exponential random variable, say T, where T ] be a function of CDF A(x; µ) of a random variable, say X, satisfying the following three conditions.
According to (Alzaatreh et al. 24 ) the CDF of T-X family defined by where, K[A(x; µ)] satisfies the conditions (I)-(III); see (Alzaatreh et al. 24 ).The PDF Y (x; µ) of T-X distribution, associated with Eq. ( 5) is as follow Now, by using m(t) = e −t as the PDF of exponential distribution with rate parameter (α = 1) and setting the upper limit K[A(x; µ)] = − log e φA(x;µ) 2 − e φ A(x; µ) 2 and lower limit b 1 = 0 in Eq. ( 5), we get the CDF Y (x; φ, µ) of the NGEP-X family, which is given by where, A(x; µ) is the CDF of any sub-model which may be depending on µ ∈ R .To prove that whether the CDF Y (x; φ, µ) is an exact CDF or not, we have the following two proposition.

Basic mathematical properties
In the present section, we derive some mathematical properties of the NGEP-X family of distributions.These properties include the QF (quantile function), moments, and MGF (moment-generating function), order statistic, and Residual and Reverse Residual life.

Quantile function
The QF is also called the inverse of the CDF and is generally used for generating RNs (random numbers) from a distribution.The RNs are usually used for simulation purposes to evaluate the performances of the estimators (or estimation method).Later in section "Simulation", we have implemented this method (i.e., inverse  distribution function) for generating RNs from the NGEP-Wei distribution.For the NGEP-Wei distributions, the QF is given by where, 0 < u < 1 and u is the solution of the above expression.The expression can be used to generate random samples or random numbers from any special model of the NGEP-X family of distributions.

rth moment
The moment is an important and useful statistical tool to obtain certain characteristics and features of any model.These characteristics are known as (i) CT (central tendency): which deals with the mean point of a distribution, (ii) MD (Measure of dispersion): which take care of the variance of a model (or measure dispersion among the data), (iii) skewness: which describe the tail behaviour of the model, and (iv) kurtosis: which helps in studying the peakiness of the distribution.For the proposed method, the r th moment expressed by µ ′ r , is derived as Using Eq. ( 8) in Eq. ( 17), we have Next, using the exponential series in Eq. ( 18), we get x a(x; µ)A(x; µ) 2m+1 dx.Furthermore, a simple general expression for the MGF of the NGEP-X random variable X, say M x (t) , is derived as

Order statistics
In distribution theory, OS is a very crucial importance.They make their appearance (or role) in the reliability analysis, problems of estimation theory and life testing in a number of ways.They can characterize the lifetime of elements or elements of a reliability system.
Let X 1 , X 2 , .. ., X k be a random set of observations of size k chosen from NGEP-X family with CDF Y (x; φ, µ) and PDF y(x; φ, µ) given by ( 7) and ( 8), respectively.Then the DF (density function) of y r:k (x) is given by " We express the 1st order statistic as X 1:k = min (X 1 , X 2 , .. ., X k ) and the kth order statistic as On using Eq. ( 21) into Eq.( 20), we get Using Eqs. ( 7) and ( 8), in Eq. ( 22), we obtain the DF of y r:k (x).

Residual and reverse residual lifetime
The RL (residual lifetime) and RRL (reverse residual lifetime) offer broader application (or characteristics) in risk management, actuarial measures, biometry, and survival analysis.The RL of the NGEP-X family with a random variable X, say R (X) (t) is defined as In addition to the RL, we also obtain the RRL of the NGEP-X family of distributions.The RRL, say R (X) (t) is given by

Estimation, experiment and simulation
This section provides a detailed description of the maximum likelihood estimation implemented for obtaining the parameter estimates of the proposed family of distributions.Furthermore, we also conduct a comprehensive Monte Carlo simulation study to assess the performance of the estimators (or the estimation method).

Maximum likelihood estimation
Several methods of estimation are proposed for obtaining the parameter estimates in the various studies.MLE (Maximum likelihood estimation) is one of the most frequently used methods of estimation.This method furnishes estimators with several important properties and can be used in the construction of confidence intervals as well as other tests for checking statistical significance.For further details about MLEs, please see 19 .This subsection provides a discussion on the MLEs approach for obtaining the parameter estimates of the NGEP-Wei distributions.

Real life application to coved-19 data sets
Here, we consider two applications from COVID-19 data sets to illustrate the fitting power of the NGEP-Wei distribution.We applied the NGEP-Wei distribution on both data sets and compared its flexibility (or fitting power) with the other rival distributions.The rival distributions of the NGEP-Wei distribution are presented in the following Table 3.The SFs of these rival distributions are the following: Next, after selecting the rival distributions, we consider certain statistical criteria (i.e., analytical goodness of fit measures) to find out (i.e., fitting power of the competing distributions) the best-suited distribution for the considered COVID-19 data sets.The mathematical formulas (or expression) of this goodness of fits measures (or statistical criteria) are given by • The CRMS (Cramer-von-Misses) • The ANDR (Anderson-Darling) • The KS (Kolmogorov-Smirnov) • The AINC (Akaike information criteria) • The BINC (Bayesian information criteria) www.nature.com/scientificreports/ • The HQINC (Hannan-Quinn information criteria) In the above expressions of the decision criteria, L(�) is the MLF (maximized likelihood function) evaluated at MLEs, n represents the sample size and p represents the number of parameters to be estimated in the model.For the NGEP-Wei distribution and Rival distributions, the values of MLEs and the above decision tools (i.e., CRMS, ANDR, KS, AINC, BINC, CAINC, and HQINC) are computed by using R-software with "method = Nelder-Mead" algorithm.In general, among the above-applied distributions to each data set, a distribution with the lowest values of the above goodness of fit measures represents best-suited distribution for the data.

Analysis of first COVID-19 data sets
The first data set (onward signified by DAST 1) consists of 36 observations and daily new death cases (due to COVID-19) recorded from the period of 10 April to 15 May 2020 in the country of Canada.The data can also be accessible via the link [https:// covid 19.who.int/].For mor details about the DS 1, we refer to Almetwally et al. 31 , and Xin et al. 32  Corresponding to DATS 1, the numerical values of MLEs along with standard errors enclosed in parenthesis of the NFEP-Wei and rival distributions (i.e., φMLE , αMLE , δMLE , âMLE , bMLE ) are recorded in Table 4. Furthermore, the numerical values of the goodness of fit measures of the NGEP-Wei and other competitive rival distributions are recorded in Table 5.According to the model selection criteria (goodness of fit measures) in Table 5, the NGEP-Wei distribution provides the best-suited fit with the minimum value of CRMS, ANDR, KS, AINC, BINC, CAINC, and HQINC as compared with rival distributions to the Canada COVID-19 dataset (DATS 1).In other words, based on model selection criteria, we can say that the NGEP-Wei distribution attains reasonable (or satisfactory) fit, which is not sufficiently (or adequately) fitted by the other rival distributions (i.e., APTra-Wei, NRLog-Wei, Kumar-Wei, Wei, MO-NH, and GAP-Wei).Consequently, the NGEP-Wei distribution provides a valuable fit to the DATS 1. Except for the numerical illustration (or comparison) of the NFEP-Wei distribution and other rival distributions, we also presented a visual illustration of the NGEP-Wei distribution.For visual illustration, we plotted the profiles of the log-likelihood function of the φMLE , αMLE , and δMLE in Fig.   in Fig. 5, we clearly see that the point estimated parameter values of the NGEP-Wei distribution are the maxima.Similarly, the PDF, CDF, SF, PP-plot and QQ-plot for the NGEP-Wei distribution are visualized in Fig. 6.From the graphical illustration in Fig. 6, we can also see that the respective red curve lines of the NGEP-Wei distribution are more close fit to the corresponding empirical objects.

Analysis of second COVID-19 data sets
The second data set (onward signified by DAST 2) consists of 108 observations.The DAST 2 represents COVID-19 mortality rates and belonging to Mexico for 108 days.This DAST 2 was recently used by Almongy et al. 33 and suggested a new extended Rayleigh distribution.Observations on the mortality rates were recorded from the period of March 4, 2020 to July 20, 2020.The DAST 2 is: DAST 2 = {8.826,6.105, 9.391, 14.962, 10.383, 7.267,  13.220, 16.498, 11.665, 6.015, 10.855, 6.122, 6.656, 3.440, 5.854, 10.685, 10.035, 5.242, 4.344, 5.143, 7.630, 14.604,  7.903, 6.370, 3.537, 6.327, 4.730, 3.215, 9.284, 12.878, 8.813, 10.043, 7.260, 5.985 , 6.412, 3.395, 4.424    www.nature.com/scientificreports/Corresponding to DATS 1, the numerical values of MLEs along with standard errors enclosed in parenthesis of the NGEP-Wei and rival distributions (i.e., φMLE , αMLE , δMLE , âMLE , bMLE ) are recorded in Table 6.Further- more, the numerical values of the goodness of fit measures of the NGEP-Wei and other competitive rival distributions are recorded in Table 7.According to the model selection criteria (goodness of fit measures) in Table 7, the NGEP-Wei distribution provides the best-suited fit with the minimum value of CRMS, ANDR, KS, AINC, BINC, CAINC, and HQINC as compared with rival distributions to the Mexico COVID-19 dataset (DATS 2).In other words, based on model selection criteria, we can say that the NGEP-Wei distribution attains reasonable (or satisfactory) fit, which is not sufficiently (or adequately) fitted by the other rival distributions (i.e., APTra-Wei, NRLog-Wei, Kumar-Wei, Wei, MO-NH, and GAP-Wei).Except for the numerical illustration (or comparison) of the NGEP-Wei distribution and other rival distributions, we also presented a visual illustration of the NGEP-Wei distribution.For visual illustration, we plotted the profiles of the log-likelihood function of φMLE , αMLE , and δMLE in Fig. 8.The plots in Fig. 8, we clearly see that the point estimated parameter values of the NGEP-Wei distribution are the maxima.Similarly, the empirical PDF, CDF, SF, PP-plot and QQ-plot are visualized for the NGEP-Wei distribution in Fig. 9. From the graphical illustration in Fig. 9, we can again see that the respective red curve lines of the NGEP-Wei distribution are more close fit to the corresponding empirical objects.

Concluding remarks
In this article, we presented a novel generator called a "Novel Generalized Exponent Power-X" family of distributions or in short NGEP-X family.A special sub case of the proposed class by employing the Weibull distribution as a baseline distribution is derived.The special sub-case is named as a Novel Generalized Exponent Power Weibull distribution (NGEP-Wei for short).The density function of the derived model is positively skewed, negatively skewed as well as symmetrical depending on parameter values.Moreover, the hazard function can be monotonically increasing, decreasing, unimodal, and bathtub-shaped.General expressions, for different statistical properties of the proposed class (NGEP-X) have been derived including quantile function, moments, moments generating function, order statistics, residual and reverse residual of lifetime.The maximum Likelihood Estimation method has been used for estimating the model parameters.In addition, a comprehensive MCSS (or

Figure 6 .
Figure 6.The empirical PDF, CDF, SF, PP-plot and QQ-plot of the NGEP-Wei distribution for DATS 1.
simulation) is carried out to assess the performance of the estimators of the proposed model.To prove the efficacy (or fitting power over other classical distributions) of the proposed family of distribution (NGEP-X) based on an NGEP-Wei distribution, we considered two data sets of COVID-19 mortality rates related to the countries of Mexico and Canada.Based on numerical illustration, it is observed that the proposed work outperforms then other widely used existing distributions.For future works, many researchers can use our proposed method to develop new extensions of the existing distributions such as a Novel Generalized Exponent Power Lomax, a Novel Generalized Exponent Power Inverse Lomax, a Novel Generalized Exponent Power Pareto, and a Novel Generalized Exponent Power Lindley that are powerful for representing and predicting real-world phenomena.

Table 2 .
MCSS results for NGEP-Wei distribution with different combination of parameters values.

Table 3 .
Rival distributions of the NGEP-Wei distribution.

Table 4 .
Estimated MLEs values along with standard errors in parenthesis for DATS 1.

Table 5 .
The analytical measures of the NGEP-Wei and others competitive models for DATS 1.

Table 6 .
Estimated MLEs values along with standard errors in parenthesis for DATS 2.